Solving nonlinear parabolic equations under nonlocal conditions by a nonlocal boundary shape function and splitting-linearizing method

In the article, we solve a nonlinear parabolic type partial differential equation (PDE) subject to non-separated and nonlocal conditions. First, a nonlocal boundary shape function (NLBSF) is derived to satisfy the initial condition and two nonlocal conditions. In the NLBSF, upon letting the free function be the Pascal polynomials the new bases can be created, which automatically fulfill all the conditions specified. The solution is then expanded in terms of these bases. Collocating points inside the space-time domain to satisfy the nonlinear PDE and in conjunction with a novel splitting-linearizing technique, quite accurate solution of the nonlocal and nonlinear parabolic equation can be achieved very fast. The numerical examples are given which confirm the high accuracy and efficiency of the proposed iterative method.